Preparation ofXAFS Samples

Grant BunkerProfessor of Physics

BCPS Department/CSRRIIllinois Institute of Technology

AcknowledgementsEd Stern, Dale Sayers, Farrel Lytle, Steve Heald, Tim Elam, Bruce Bunker and other early members of the UW XAFS group

Firouzeh Tannazi (IIT/BCPS) (recent results on fluorescence in complex materials)

Suggested References:

Steve Heald’s article “designing an EXAFS experiment” in Koningsberger and Prins andearly work cited therein (e.g. Stern and Lu)

Rob Scarrow’s notes posted athttp://cars9.uchicago.edu/xafs/NSLS_2002/

Experimental ModesModes:

Transmission

Fluorescence

Electron yield

Designing the experiment requires an understanding of sample preparation methods, experimental modes, and data analysis

Comparison to theory requires stringent attention to systematic errors - experimental errors don’t cancel out with standard

Simplest XAFS measurement

Measure relative x-ray flux transmitted through homogeneous sample

Transmission

uniform sampleUniform, homogeneous sample:

I

I0= exp(!µ(E)x)

x is the sample thickness

µ(E) is the linear x-ray absorption coe!cientat x-ray energy E

Decreases roughly as 1/E3 between absorp-tion edges

Absorption Length

distance over which x-ray intensity decreases by factor 1/e ~ 37%

sets the fundamental length scale for choosing sample thickness, particle size, and sample homogeneity

You should calculate it when designing experiments

“Absorption Length”! 1/µ

Absorption CoefficientSingle substance:

µ = !"

! is the density; " is the cross section.

If the units of ! are g/cm3 the cross sectionis in cm2/g.

If the units of ! are atoms/cm3 the crosssection is in cm2/atom.

1barn = 10!24cm2.

Cross section

Definition of ”cross section” !:

R[photons

s] = ![

photons

s ! cm2 ]!![cm2

atom]!N [atom],

alternatively

R[photons

s] = ![

photons

s ! cm2 ] ! ![cm2

g] ! M [g]

Interaction between a beam of particles (photons) and a target

Sources of Cross Section DataS. Brennan and P.L. Cowan, Rev. Sci. Instrum, vol 63, p.850 (1992).

C. T. Chantler, J. Phys. Chem. Ref. Data 24, 71 (1995) http://physics.nist.gov/PhysRefData/FFast/html/form.html

W.T. Elam, B.Ravel, and J.R. Sieber, Radiat. Phys. Chem. v.63 (2002) pp 121-128.

B. L. Henke, E. M. Gullikson, and J. C. Davis, Atomic Data and Nuclear Data Tables Vol. 54 No. 2 (1993). http://www-cxro.lbl.gov/optical_constants/atten2.html http://www-cxro.lbl.gov/optical_constants/

J.H. Hubbell, Photon Mass Attenuation and Energy-Absorption Coefficients from 1 keV to 20 MeV, Int. J. Appl. Radiat. Isot. 33, 1269-1290 (1982)

W.H. McMaster et al. Compilation of X-ray Cross Sections. Lawrence Radiation Laboratory Report UCRL-50174, National Bureau of Standards, pub. (1969).http://www.csrri.iit.edu/mucal.htmlhttp://www.csrri.iit.edu/periodic-table.html

compoundsAbsorption coe!cient approximately given by

µ !!

i

!i"i = !M!

i

mi

M"i = !N

!

i

ni

N"i

where !M is the mass density of the mate-rial as a whole, !N is the number density ofthe material as a whole, and mi/M and ni/N

are the mass fraction and number fractionof element i.

Fe3O4 (magnetite) at 7.2 KeV;http://www.csrri.iit.edu/periodic-table.html

density 5.2 gcm3

MW=3 ! 55.9 gmol + 4 ! 16.0 g

mol = 231.7 gmol

!Fe = 393.5cm2

g ; MFe = 55.9 gmol;

fFe = 55.9/231.7 = .724;

!O = 15.0 cm2

g ; MO = 16.0 gmol;

fO = 16.0/231.7 = .276;

µ = 5.2 gcm3(.724!393.5cm2

g + .276!15.0cm2

g )= 1503/cm = .15/micronAbsorption Length = 1µm/.15 = 6.7 microns

Sample Calculation

Even if you don’t know the density exactly you can estimate it from something similar. It’s probably between 2 and 8 g/cm^3

Transmissionnonuniform sample

Nonuniform Sample:

Characterized by thickness distribution P (x)

µxe!(E) = ! ln! "

0P (x) exp (!µ(E)x)dx

= !""

n=1

Cn(!µ)n

n!,

where Cn are the cumulants of the thick-ness distribution (C1 = x̄, C2 = mean squarewidth, etc.)

ref gb dissertation 1984

A Gaussian distribution of width ! has

µxe!(E) = µx̄! µ2!2/2

What’s theproblem withnonuniformsamples?

Effect of Gaussian thickness variation

P (x)

µxe! and µx̄ vs µ

µxe! vs µ

P (x)

µxe! and µx̄ vs µ

µxe! vs µ

P (x)

µxe! and µx̄ vs µ

µxe!! vs µ

P (x)

µxe! and µx̄ vs µ

µxe!! vs µ

! = 0.1

! = 0.3

P (x)

µxe! and µx̄ vs µ

µxe!! vs µ

! = 0.1

! = 0.3

P (x)

µxe! and µx̄ vs µ

µxe!! vs µ

! = 0.1

! = 0.3

P (x) =1

!!

(2")exp (

"(x " x̄)2

2!2 )

1 2 3 4

10

20

30

1 2 3 4 5

1

2

3

4

51 2 3 4 5

0.5

0.6

0.7

0.8

0.9

Effect of leakage/harmonics

Leakage (zero thickness) fraction a, togetherwith gaussian variation in thickness centeredon x0 with width !:

P (x) = a"(x) + (1! a)1

!"

2#exp (

!(x! x̄)2

2!2 )

µxe!(E) = ! ln (a + (1! a) exp (!µx0 + µ2!2/2))

Effect of pinholes (leakage) or harmonics

P (x)

µxe! and µx̄ vs µ

µxe! vs µ

P (x)

µxe! and µx̄ vs µ

µxe! vs µ

P (x)

µxe! and µx̄ vs µ

µxe!! vs µ

a = .05

a = .02

a = .05

a = .02

Leakage (zero thickness) fraction a, togetherwith gaussian variation in thickness centeredon x0 with width !:

P (x) = a"(x) + (1! a)1

!"

2#exp (

!(x! x̄)2

2!2 )

µxe!(E) = ! ln (a + (1! a) exp (!µx0 + µ2!2/2))

MnO 10 micron thick ~2 absorption lengths

leakage varied from 0% to 10%

Edge jump is reduced

EXAFS amplitudes are reduced

white line height compressed

thickness effects distort both XANES and EXAFS - screw up fits and integrals of peak areas

If you are fitting XANES spectra, watch out for these distortions

Effect of Leakage on spectra

Thickness effects alwaysreduce EXAFS Amplitudes

from http://gbxafs.iit.edu/training/thickness_effects.pdf

Text

Thickness distribution is a sum of gaussiansof weight an, thickness xn, and width !n:

P (x) =!

n

an

!n!

2"exp (

"(x" xn)2

2!2n

)

µxe!(E) = " ln (!

nan exp ("µxn + µ2!2

n/2))

This expression can be used to estimatethe effect of thickness variations

Simple model of thickness distribution

Example - Layers of spheres

square lattice - holes in one layer covered by spheres in next layer

ThicknessDistribution

Transmission -Summary

Samples in transmission should be made uniform on a scale determined by the absorption length of the material

Absorption length should be calculated when you’re designing experiments and preparing samples

When to choose TransmissionYou need to get x-rays through the sample

Total thickness should be kept below <2-3 absorption lengths including substrates to minimize thickness effects

“beam hardening” - choose fill-gases of back ion chamber to minimize absorption of harmonics; get rid of harmonics by monochromator detuning, harmonic rejection mirrors, etc.

Element of interest must be concentrated enough to get a decent edge jump (> 0.1 absorption length)

Pinholes and large thickness variations should be minimized

If you can’t make a good transmission sample, consider using fluorescence or electron yield

Fluorescence Radiation in the Homogeneous Slab Model

• Probability the photon penetrates to a depth x in the sample

• and that is absorbed by the element i in a layer of thickness dx

• and as a consequence it emits with probability ε a fluorescence photon of energy Ef

• which escapes the sample and is radiated into the detector • Thin Sample

Fluorescence samplesThin concentrated limit simple

Thick dilute limit simple

Thick concentrated requires numerical corrections(e.g. Booth and Bridges). Thickness effects can be correctedalso if necessary by regularization (Babanov et al).

Sample Requirements

Particle size must be small compared to absorption lengths of particles (not just sample average)

Can be troublesome for in situ studies

Homogeneous distribution

Flat sample surface preferred

Speciation problems

47

On the other hand when d!", i.e. the “thick limit”, the exponential term

will vanish and for thick samples we will have:

(If )thick =I0!a(

µa

sin ! )µT

sin ! + µf

sin "

(5.6)

Fluorescence vs Particle Size

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

6400 6500 6600 6700 6800 6900

Energy (ev)

If

x=0.125

x=0.25

x=0.5

x=1

x=2

x=4

x=8

Figure 5.2. Fluorescence

But in general the measured fluorescence radiation depends nonlinearily on

µa. The nonlinearity problems arise when the sample consists of concentrated species

with particle sizes that are not small compared to one absorption length. In this

case the penetration depth into the particles has an energy dependence that derives

from the absorption coe!cient of the element of interest, making the interpretation of

the results much more complicated. Fig.5.2 [21] simulates how the measured XAFS

Nonlinear distortions of the spectra depend on particle size and distribution. This affects speciation

results

Modeling Fluorescence

Monte Carlo and analytical calculations of Tannazi and Bunker

Analytical calculations build on work by Hunter and Rhodes, and Berry, Furuta and Rhodes (1972)

Computation of fluorescence radiation from arbitrarily shaped convex particles by

Monte Carlo methods

Incomingphoton

direction

Outgoingfluorescence

photon direction

€

r r

€

ˆ n 0

€

ˆ n 1Particle

€

d0€

d1

The probability of penetrating to an

arbitrary position within the particle is calculated.

This probability is averaged over the whole particle by Monte Carlo

integration.

CuboidalParticles(stereo)

Calculate theprobabilityas functionof mu and

mu_f

Different Orientations

d0,d1 maps

Text

Even the particle orientation mattersif particles too large

Cumulant Coefficients

• The log of the mean probability can be expanded as a power series in both µ and µf. The coefficients are related to the cumulants of the (2D) distribution of distances d0,d1.The main point is that the probabilities for a given shape of particle (and theta, phi) can be parameterized by a handful of numbers, the coefficients.

Other shapestabulated in

Firouzeh Tannazidissertation

DilutionEven if the sample is dilute on average you may not be really in the dilute limit

Each individual particle must be small enough, otherwise you will get distorted spectra

Don’t just mix up your particles with a filler and assume it’s dilute. Make them small first.

Hunter and Rhodes ModelContinuous Size Distribution

• This model is a generalization of BFR model that has been developed by the authors above (1972), and is a formulation for continuous size distribution.

• In this approach we need to define a particle size distribution function:

• Where amin and amax are the smallest and larges particle size in the sample and a is the particle size that is a variable in this approach.

• The probabilities term compare to the BFR model are defined as differential probabilities here: Pf(a,a’) da.

• For calculation average transmission or fluorescence radiation through I layers we need to consider three different cases: No overlap in all layers, Some overlap and, total overlap.

• The total Fluorescence radiation is obtained from this formula, which we are adapting to XAFS.

Io If

Fluorescent Particle

d

{(i+1)th Layer

d'

Non-fluorescent Particle

€

f (a)daamin

amax∫ = 1

HR Model

HR Model

nonlinearcompressionof spectra

€

η→1

€

Cf →1

Io If

Fluorescent Particle

d

{(i+1)th Layer

d'

Non-fluorescent Particle

€

f (x) =δ (x − a )

Comparison between BFR, HR, and Slab Model

All 3 modelsagree in theappropriate

limits

Summary - FluorescenceParticle size effects are important in fluorescence as well as transmission

The homogeneous slab model is not always suitable but other models have been developed

If the particles are not sufficiently small, their shape, orientation, and distribution can affect the spectra in ways that can influence results

Particularly important for XANES and speciation fitting

Electron yield detectionsample placed atop and electrically connected to cathode of helium filled ion chamber

electrons ejected from surface of sample ionize helium - their number is proportional to absorption coefficient

The current is collected as the signal just like an ionization chamber

Surface sensitivity of electron detection eliminates self-absorption problems but does not sample bulk material

Sample Inhomogeneities

Importance of inhomogeneity also can depend on spatial structure in beam

bend magnets and wiggler beams usually fairly homogeneous

Undulator beams trickier because beam partially coherent

coherence effects can result in spatial microstructure in beam at micron scales

Beam Stability

Samples must not have spatial structure on the same length scales as x-ray beam. Change the sample, or change the beam.

Foils and films often make good transmission samples

stack multiple films if possible to minimize through holes

check for thickness effects (rotate sample)

Foils and films can make good thin concentrated or thick dilute samples in fluorescence

Consider grazing incidence fluorescence to reduce background and enhance surface sensitivity

SolutionsUsually solutions are naturally homogeneous

Can make good transmission samples if concentrated

Usually make good fluorescence samples (~1 millimolar and 100 ppm are routine), lower concentrations feasible

They can become inhomogeneous during experiment

phase separation

radiation damage can cause precipitation (e.g. protein solutions)

photolysis of water makes holes in intense beams

suspensions/pastes can be inhomogeneous

Particulate samplesFirst calculate the absorption length for the material

prepare particles that are considerably smaller than one absorption length of their material, at an energy above the edge

Many materials require micron scale particles for accurate results

Distribute the particles uniformly over the sample cross sectional area by dilution or coating

Making Fine Particles

During synthesis -> choose conditions to make small particles

Grinding and separating

sample must not change during grinding (e.g. heating)

For XAFS can’t use standard methods (e.g. heating in furnace and fluxing) from x-ray spectrometry, because chemical state matters

Have to prevent aggregation back into larger particles

Grinding MaterialsDilute samples - have to prevent contamination

mortar and pestle (porcelain or agate (a form of quartz))

inexpensive small volume ball mill (e.g. “wig-l-bug” ~$700 US)

agate vials available

disposable plastic vials for mixing

standard for infrared spectrometry

Frisch mill (several thousands of $K US) e.g. MiniMill 2 Panalytical, Gilson Micromill

Sieves3” ASTM sieves work well

Screen out larger particles with coarse mesh (100 mesh)

pass along to next finer sieve

Sieve stack and shaker

Usually can do well with 100, 200, 325, 400, 500, 635 mesh

Still only guarantees 20 micron particles, too big for many samples

100 mesh 150 µm

115 mesh 125 µm

150 mesh 106 µm

170 mesh 90 µm

200 mesh 75 µm

250 mesh 63 µm

270 mesh 53 µm

325 mesh 45 µm

400 mesh 38 µm

450 mesh 32 µm

500 mesh 25 µm

635 mesh 20 µm

size (microns)~15000/mesh

sedimentationDrag force on a spherical particle of radiusR moving at velocity v in fluid of viscosity!: F = 6"!Rv. Particles of density # will fallthrough the fluid of density #0 at a speedin which the particle’s weight, less the buoy-ancy force, equals the drag force:

(#! #0)4

3"R3g = 6"!Rv.

If the height of the fluid in the container ish, the time t that it would take all particlesof radius R to fall to the bottom would thenbe:

t =9

5

!h

(#! #0)gR2

Selecting yet smaller particles

Sedimentation cont’dBy knowing the densities of the material and the liquid (e.g. acetone), and the viscosity, the fluid height, and the required particle radius, you calculate the time.

Mix it in, stir it up

wait the required time

decant the supernatant with a pipette

dry particles in a dish in a fume hood

Must have non-reactive liquid

worked example

Example: MnO in acetone at 20Cviscosity of acetone :! = 0.0032 Poise= 0.0032 g/(cm ! s)R = 1µm = 3 ! 10"4cmdensity of acetone: "0 = 0.79 g

cm3

density of MnO: " = 5.4 gcm3

h = 5cmg = 980cm/s2

# t = 638 seconds.

Assembling SamplesA) Mix uniformly (use wig-l-bug) into filler or binder

nonreactive, devoid of the element you are measuring, made of a not-too-absorbant substance e.g. Boron Nitride, polyvinyl alcohol, corn starch

Place into a sample cell (x-ray transparent windows e.g kapton or polypropylene), or press to make pellet

B) Apply uniform coating to adhesive tape

Scotch Magic Transparent Tape (clean, low absorption)

use multiple layers to cover gaps between particles

watch for brush marks/striations

C) Make a “paint” (e.g. Duco cement thinned with acetone)

Checking the samplecheck composition by spectroscopy or diffraction

visually check for homogeneity

caveat: x-ray vs optical absorption lengths

tests at beamline: move and rotate sample

digital microscope (Olympus Mic-D $800 US)

particle size analysis (Image/J free)

If you have nice instruments like a scanning electron microscope or light scattering particle size analyzer, don’t hesitate to use them

Preferred OrientationIf your sample is polycrystalline, it may orient in non-random way if applied to a substrate

since the x-ray beams are polarized this can introduce an unexpected sample orientation dependence

Test by changing sample orientation

Magic-angle spinning to eliminate effect

Control samplesIn fluorescence measurements always measure a blank sample without the element of interest under the same conditions as your real sample

Many materials have elements in them that you wouldn’t expect that can introduce spurious signals

Most aluminum alloys have transition metals

Watch out for impurities in adhesive films

Fluorescence from sample environment excited by scattered x-rays and higher energy fluorescence

HALOHarmonics

get rid of them

Alignment

beam should only see uniform sample

Linearity

detectors and electronics must linear

Offsets

subtract dark currents regularly

ConclusionCalculate the absorption coefficients so you know what you are dealing with at the energies you care about. Think like an x-ray. Know what to expect.

Absorption coefficients increase dramatically at low energies ~ 1/E^3. Know the penetration depths.

Make particles small enough and samples homogeneous

Check experimentally for thickness, particle size, and self absorption effects

Choose materials so the sample is not reactive with sample cell or windows.